It seems that you have a conviction for a Class B Misdemeanor Driving While Intoxicated on your record. Please check and make sure with the local County Clerk.

If so, then this does not prevent you from OWNING a handgun. However, it does prevent you from obtaining a Concealed Handgun License because the conviction is still within the last 5 years. After 5 years is up, you may apply for your CHL if you also are ellgible under the factors listed below.

There are other factors or laws that may prevent you from OWNING a gun, but not based solely on a conviction of a Class B Misdemeanor DWI. For more information on a CHL, google the Texas Concealed Handgun Laws.

GC §411.172. ELIGIBILITY. (a) A person is eligible for a license to
carry a concealed handgun if the person:
(1) is a legal resident of this state for the six-month period preceding
the date of application under this subchapter or is otherwise eligible for
a license under Section 411.173(a);
(2) is at least 21 years of age;
(3) has not been convicted of a felony;
(4) is not charged with the commission of a Class A or Class B misdemeanor
or equivalent offense, or of an offense under Section 42.01,
Penal Code, or equivalent offense, or of a felony under an information
or indictment;
(5) is not a fugitive from justice for a felony or a Class A or Class B
misdemeanor or equivalent offense;
(6) is not a chemically dependent person;
(7) is not incapable of exercising sound judgment with respect to the
proper use and storage of a handgun;
(8) has not, in the five years preceding the date of application, been
convicted of a Class A or Class B misdemeanor or equivalent offense or
of an offense under Section 42.01, Penal Code, or equivalent offense;
(9) is fully qualified under applicable federal and state law to purchase
a handgun;
(10) has not been finally determined to be delinquent in making a
child support payment administered or collected by the attorney general;
(11) has not been finally determined to be delinquent in the payment
of a tax or other money collected by the comptroller, the tax collector of
a political subdivision of the state, or any agency or subdivision of the
state;
(12) is not currently restricted under a court protective order or subject
to a restraining order affecting the spousal relationship, other than
a restraining order solely affecting property interests;
(13) has not, in the 10 years preceding the date of application, been
adjudicated as having engaged in delinquent conduct violating a penal
law of the grade of felony; and
(14) has not made any material misrepresentation, or failed to disclose
any material fact, in an application submitted pursuant to Section
411.174.